Qualitative Behavior for Fourth-Order Nonlinear Differential Equations

Omar Bazighifan; Elmetwally Elabbasy; Osama Moaaz

doi:doi:10.11648/j.engmath.20180202.12

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Qualitative Behavior for Fourth-Order Nonlinear Differential Equations

Omar Bazighifan, Elmetwally Elabbasy, Osama Moaaz

Published in Engineering Mathematics (Volume 2, Issue 2)

Received: 1 October 2018 Accepted: 16 October 2018 Published: 5 November 2018

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Abstract

In this paper, new an oscillation criterion for a class of fourth-order nonlinear delay differential equations are established by using the double generalized Riccatti substitutions. Recently, there has been increasing interest related to the theory of delay differential equations (DDEs), this has been attributed to the important of understanding of application in delay differential equations. Recent applications that include delay differential equations continue to appear with increasing frequency in the modeling of diverse phenomena in physics, biology, ecology, and physiology. So, other authors have been attracted to finding the solutions of the differential equations or deducing important characteristics of them has received the attention of many authors. The solution to this equation is important in order to understand these issues and phenomena, or at least to know the characteristics of the solutions to these equations. But, differential equations such as those used to solve real life problems may not necessarily be directly solvable, i.e., do not have closed form solutions. The main objective of this work is to provide an opportunity to study the new trends and analytical insights of the delay differential equations, existence and uniqueness of the solutions, boundedness and persistence, oscillatory behavior of the solutions, stability and bifurcation analysis, parameter estimations and sensitivity analysis, and numerical investigations of solutions. One objective of our paper is to further simplify and complement some well-known results which were published recently in the literature. An illustrative example is included.

Published in	Engineering Mathematics (Volume 2, Issue 2)
DOI	10.11648/j.engmath.20180202.12
Page(s)	63-67
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Oscillation, Fourth-Order, Delay Differential Equations

References

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[2]	R. Agarwal, S. Grace and D. O'Regan, Oscillation theory for difference and functional differential equations, Kluwer Acad. Publ., Dordrecht (2000).
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[5]	O. Bazighifan, Oscillation criteria for nonlinear delay differential equation, Lambert Academic Publishing, Germany, (2017).
[6]	O. Bazighifan, Oscillatory behavior of higher-order delay differential equations, General Letters in Mathematics, 2 (2017), 105-110.
[7]	Džurina and Jadlovskă, Oscillation theorems for fourth order delay differential equations with anegative middle term, Math. Meth. Appl. Sci., 42 (2017), 1-17.
[8]	E. M. Elabbasy, O. Moaaz and O. Bazighifan, Oscillation Solution for Higher-Order Delay Differential Equations, Journal of King Abdulaziz University, 29 (2017), 45-52.
[9]	E. M. Elabbasy, O. Moaaz and O. Bazighifan, Oscillation of Fourth-Order Advanced Differential Equations, Journal of Modern Science and Engineering, 3 (2017), 64-71.
[10]	E. M. Elabbasy, O. Moaaz and O. Bazighifan, Oscillation Criteria for Fourth-Order Nonlinear Differential Equations, International Journal of Modern Mathematical Sciences, 15 (2017), 50-57.
[11]	S. Grace and B. Lalli, Oscillation theorems for nth order nonlinear differential equations with deviating arguments, Proc. Am. Math. Soc., 90 (1984), 65-70.
[12]	S. Grace, R. Agarwal and J. Graef, Oscillation theorems for fourth order functional differential equations, J. Appl. Math. Comput., 30 (2009), 75--88).
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[17]	O. Moaaz, E. M. Elabbasy and O. Bazighifan, On the asymptotic behavior of fourth-order functional differential equations, Adv. Difference Equ., 261 (2017), 1-13.
[18]	O. Moaaz, Oscillation properties of some differential equations, Lambert Academic Publishing, Germany, (2017).
[19]	O. Moaaz, E. M. Elabbasy and E. Shaaban, Some oscillation criteria for a class of third order nonlinear damped differential equations, International Journal of Modern Mathematical Sciences, 15 (2017), 206-218.
[20]	C. Tunc and O. Bazighifan, Some new oscillation criteria for fourth-order neutral differential equations with distributed delay, Electronic Journal of Mathematical Analysis and Applications, 7 (2019), 235-241.
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Omar Bazighifan, Elmetwally Elabbasy, Osama Moaaz. (2018). Qualitative Behavior for Fourth-Order Nonlinear Differential Equations. Engineering Mathematics, 2(2), 63-67. https://doi.org/10.11648/j.engmath.20180202.12

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ACS Style

Omar Bazighifan; Elmetwally Elabbasy; Osama Moaaz. Qualitative Behavior for Fourth-Order Nonlinear Differential Equations. Eng. Math. 2018, 2(2), 63-67. doi: 10.11648/j.engmath.20180202.12

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AMA Style

Omar Bazighifan, Elmetwally Elabbasy, Osama Moaaz. Qualitative Behavior for Fourth-Order Nonlinear Differential Equations. Eng Math. 2018;2(2):63-67. doi: 10.11648/j.engmath.20180202.12

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@article{10.11648/j.engmath.20180202.12,
  author = {Omar Bazighifan and Elmetwally Elabbasy and Osama Moaaz},
  title = {Qualitative Behavior for Fourth-Order Nonlinear Differential Equations},
  journal = {Engineering Mathematics},
  volume = {2},
  number = {2},
  pages = {63-67},
  doi = {10.11648/j.engmath.20180202.12},
  url = {https://doi.org/10.11648/j.engmath.20180202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20180202.12},
  abstract = {In this paper, new an oscillation criterion for a class of fourth-order nonlinear delay differential equations are established by using the double generalized Riccatti substitutions. Recently, there has been increasing interest related to the theory of delay differential equations (DDEs), this has been attributed to the important of understanding of application in delay differential equations. Recent applications that include delay differential equations continue to appear with increasing frequency in the modeling of diverse phenomena in physics, biology, ecology, and physiology. So, other authors have been attracted to finding the solutions of the differential equations or deducing important characteristics of them has received the attention of many authors. The solution to this equation is important in order to understand these issues and phenomena, or at least to know the characteristics of the solutions to these equations. But, differential equations such as those used to solve real life problems may not necessarily be directly solvable, i.e., do not have closed form solutions. The main objective of this work is to provide an opportunity to study the new trends and analytical insights of the delay differential equations, existence and uniqueness of the solutions, boundedness and persistence, oscillatory behavior of the solutions, stability and bifurcation analysis, parameter estimations and sensitivity analysis, and numerical investigations of solutions. One objective of our paper is to further simplify and complement some well-known results which were published recently in the literature. An illustrative example is included.},
 year = {2018}
}

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TY  - JOUR
T1  - Qualitative Behavior for Fourth-Order Nonlinear Differential Equations
AU  - Omar Bazighifan
AU  - Elmetwally Elabbasy
AU  - Osama Moaaz
Y1  - 2018/11/05
PY  - 2018
N1  - https://doi.org/10.11648/j.engmath.20180202.12
DO  - 10.11648/j.engmath.20180202.12
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 63
EP  - 67
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20180202.12
AB  - In this paper, new an oscillation criterion for a class of fourth-order nonlinear delay differential equations are established by using the double generalized Riccatti substitutions. Recently, there has been increasing interest related to the theory of delay differential equations (DDEs), this has been attributed to the important of understanding of application in delay differential equations. Recent applications that include delay differential equations continue to appear with increasing frequency in the modeling of diverse phenomena in physics, biology, ecology, and physiology. So, other authors have been attracted to finding the solutions of the differential equations or deducing important characteristics of them has received the attention of many authors. The solution to this equation is important in order to understand these issues and phenomena, or at least to know the characteristics of the solutions to these equations. But, differential equations such as those used to solve real life problems may not necessarily be directly solvable, i.e., do not have closed form solutions. The main objective of this work is to provide an opportunity to study the new trends and analytical insights of the delay differential equations, existence and uniqueness of the solutions, boundedness and persistence, oscillatory behavior of the solutions, stability and bifurcation analysis, parameter estimations and sensitivity analysis, and numerical investigations of solutions. One objective of our paper is to further simplify and complement some well-known results which were published recently in the literature. An illustrative example is included.
VL  - 2
IS  - 2
ER  -

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Author Information

Omar Bazighifan

Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout, Yemen
Elmetwally Elabbasy

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
Osama Moaaz

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

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Plain Text BibTeX RIS

APA Style

Omar Bazighifan, Elmetwally Elabbasy, Osama Moaaz. (2018). Qualitative Behavior for Fourth-Order Nonlinear Differential Equations. Engineering Mathematics, 2(2), 63-67. https://doi.org/10.11648/j.engmath.20180202.12

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ACS Style

Omar Bazighifan; Elmetwally Elabbasy; Osama Moaaz. Qualitative Behavior for Fourth-Order Nonlinear Differential Equations. Eng. Math. 2018, 2(2), 63-67. doi: 10.11648/j.engmath.20180202.12

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AMA Style

Omar Bazighifan, Elmetwally Elabbasy, Osama Moaaz. Qualitative Behavior for Fourth-Order Nonlinear Differential Equations. Eng Math. 2018;2(2):63-67. doi: 10.11648/j.engmath.20180202.12

Copy | Download

@article{10.11648/j.engmath.20180202.12,
  author = {Omar Bazighifan and Elmetwally Elabbasy and Osama Moaaz},
  title = {Qualitative Behavior for Fourth-Order Nonlinear Differential Equations},
  journal = {Engineering Mathematics},
  volume = {2},
  number = {2},
  pages = {63-67},
  doi = {10.11648/j.engmath.20180202.12},
  url = {https://doi.org/10.11648/j.engmath.20180202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20180202.12},
  abstract = {In this paper, new an oscillation criterion for a class of fourth-order nonlinear delay differential equations are established by using the double generalized Riccatti substitutions. Recently, there has been increasing interest related to the theory of delay differential equations (DDEs), this has been attributed to the important of understanding of application in delay differential equations. Recent applications that include delay differential equations continue to appear with increasing frequency in the modeling of diverse phenomena in physics, biology, ecology, and physiology. So, other authors have been attracted to finding the solutions of the differential equations or deducing important characteristics of them has received the attention of many authors. The solution to this equation is important in order to understand these issues and phenomena, or at least to know the characteristics of the solutions to these equations. But, differential equations such as those used to solve real life problems may not necessarily be directly solvable, i.e., do not have closed form solutions. The main objective of this work is to provide an opportunity to study the new trends and analytical insights of the delay differential equations, existence and uniqueness of the solutions, boundedness and persistence, oscillatory behavior of the solutions, stability and bifurcation analysis, parameter estimations and sensitivity analysis, and numerical investigations of solutions. One objective of our paper is to further simplify and complement some well-known results which were published recently in the literature. An illustrative example is included.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Qualitative Behavior for Fourth-Order Nonlinear Differential Equations
AU  - Omar Bazighifan
AU  - Elmetwally Elabbasy
AU  - Osama Moaaz
Y1  - 2018/11/05
PY  - 2018
N1  - https://doi.org/10.11648/j.engmath.20180202.12
DO  - 10.11648/j.engmath.20180202.12
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 63
EP  - 67
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20180202.12
AB  - In this paper, new an oscillation criterion for a class of fourth-order nonlinear delay differential equations are established by using the double generalized Riccatti substitutions. Recently, there has been increasing interest related to the theory of delay differential equations (DDEs), this has been attributed to the important of understanding of application in delay differential equations. Recent applications that include delay differential equations continue to appear with increasing frequency in the modeling of diverse phenomena in physics, biology, ecology, and physiology. So, other authors have been attracted to finding the solutions of the differential equations or deducing important characteristics of them has received the attention of many authors. The solution to this equation is important in order to understand these issues and phenomena, or at least to know the characteristics of the solutions to these equations. But, differential equations such as those used to solve real life problems may not necessarily be directly solvable, i.e., do not have closed form solutions. The main objective of this work is to provide an opportunity to study the new trends and analytical insights of the delay differential equations, existence and uniqueness of the solutions, boundedness and persistence, oscillatory behavior of the solutions, stability and bifurcation analysis, parameter estimations and sensitivity analysis, and numerical investigations of solutions. One objective of our paper is to further simplify and complement some well-known results which were published recently in the literature. An illustrative example is included.
VL  - 2
IS  - 2
ER  -

Copy | Download